Robust Estimation Problem Research Articles

A neural network for robust estimation and uncertainty intervals evaluation of nonlinear models

Artificial neural networks are dynamic systems consisting of highly interconnected and parallel nonlinear processing elements. Systems based on artificial neural networks have high computational rates due to the use of a massive number of simple processing elements and the high degree of connectivity between these elements. The ability of neural networks to realize some complex nonlinear function makes them attractive for system identification. This paper presents a novel approach to solve robust parameter estimation problems for nonlinear model with unknown-but-bounded errors and uncertainties. More specifically, a modified Hopfield network is developed and its internal parameters are computed using the valid-subspace technique. These parameters guarantee the network convergence to the equilibrium points. A solution for the robust estimation problem with unknown-but-bounded error corresponds to an equilibrium point of the network. Simulation results are presented as an illustration of the proposed approach.

Minimization with few violated constraints and its application in set-membership identification

Abstract We study the problem of finding an optimal solution satisfying all but k of the given n constraints. A solution is obtained along with an algorithm of the complexity min{ O(n. k d ), O(n. d k+1 ) }, where d is the dimension of the problem. We then use the results to solve successfully the problem of robust estimation in the presence of outliers in the setting of bounded error parameter identification. It is shown that the obtained estimate converges to the true but unknown parameter even in the presence of unknown outliers.

A novel approach to robust parameter estimation using neurofuzzy systems

A novel approach for solving robust parameter estimation problems is presented for processes with unknown-but-bounded errors and uncertainties. An artificial neural network is developed to calculate a membership set for model parameters. Techniques of fuzzy logic control lead the network to its equilibrium points. Simulated examples are presented as an illustration of the proposed technique. The result represent a significant improvement over previously proposed methods.

Robust Bayesian estimation in a normal model with asymmetric loss function

The problem of robust Bayesian estimation in a normal model with asymmetric loss function (LINEX) is considered. Some uncertainty about the prior is assumed by introducing two classes of priors. The most robust and conditional Γ-minimax estimators are con

Robust discriminant analysis: Training data breakdown point

This paper discusses the robustness of discriminant analysis against contamination in the training data, the test data are assumed uncontaminated. The concept of training data breakdown point for discriminant analysis is introduced. It is quite different from the usual breakdown point in robust statistics. In the robust location parameter estimation problem, outliers are the main concern, but in discriminant analysis, not only are outliers a concern, but also inliers.

Robust estimation of parametric membership regions

This paper is concerned with the robust identification of linear models when modelling error is bounded. A modified Hopfield's neural network is used to calculate a membership set for the model parameters, with the internal parameters of the network obtained using the valid-subspace technique. These parameters can be explicitly computed to guarantee the network convergence. A solution for the robust estimation problem with an unknown-but-bounded error corresponds to an equilibrium point of the network. A comparative analysis with alternative robust estimation methods is provided to illustrate the proposed approach.

Robust filtering with missing data and a deterministic description of noise and uncertainty

The paper considers the problem of robust state estimation for the case in which some of the measurement data is missing. This problem is considered within the framework of a class of uncertain discrete-time systems with a deterministic description of noise and uncertainty. The main result is a recursive scheme for constructing an ellipsoidal state estimation set of all states consistent with the available measured output and the given noise and uncertainty description.

On robust estimation of the common scale parameter of several Pareto distributions

The problem of robust estimation of the common scale parameter of several Pareto distributions with unknown and possibly unequal shape parameters is considered. In this paper, a wide class of estimators dominating the maximum likelihood estimator (MLE) is derived under a class of convex loss functions. The problem discussed in this paper arises quite frequently in socio-economics, reliability, life testing, and survival analysis.

Robust state estimation for uncertain systems with averaged integral quadratic constraints

This paper considers a robust state estimation problem for a new class of uncertain systems. The new uncertainty class introduced in the paper involves structured uncertainties which are required to satisfy a certain averaged integral quadratic constraint. This uncertainty class combines aspects of norm-bounded and stochastic uncertainty descriptions. The solution to the robust state estimation problem is obtained by solving a parametrized Riccati differential equation of the game type.

On the unification of line processes, outlier rejection, and robust statistics with applications in early vision

The modeling of spatial discontinuities for problems such as surface recovery, segmentation, image reconstruction, and optical flow has been intensely studied in computer vision. While “line-process” models of discontinuities have received a great deal of attention, there has been recent interest in the use of robust statistical techniques to account for discontinuities. This paper unifies the two approaches. To achieve this we generalize the notion of a “line process” to that of an analog “outlier process” and show how a problem formulated in terms of outlier processes can be viewed in terms of robust statistics. We also characterize a class of robust statistical problems for which an equivalent outlier-process formulation exists and give a straightforward method for converting a robust estimation problem into an outlier-process formulation. We show how prior assumptions about the spatial structure of outliers can be expressed as constraints on the recovered analog outlier processes and how traditional continuation methods can be extended to the explicit outlier-process formulation. These results indicate that the outlier-process approach provides a general framework which subsumes the traditional line-process approaches as well as a wide class of robust estimation problems. Examples in surface reconstruction, image segmentation, and optical flow are presented to illustrate the use of outlier processes and to show how the relationship between outlier processes and robust statistics can be exploited. An appendix provides a catalog of common robust error norms and their equivalent outlier-process formulations.

Robustness properties of S-estimators of multivariate location and shape in high dimension

For the problem of robust estimation of multivariate location and shape, defining S-estimators using scale transformations of a fixed $\rho$ function regardless of the dimension, as is usually done, leads to a perverse outcome: estimators in high dimension can have a breakdown point approaching 50%, but still fail to reject as outliers points that are large distances from the main mass of points. This leads to a form of nonrobustness that has important practical consequences. In this paper, estimators are defined that improve on known S-estimators in having all of the following properties: (1) maximal breakdown for the given sample size and dimension; (2) ability completely to reject as outliers points that are far from the main mass of points; (3) convergence to good solutions with a modest amount of computation from a nonrobust starting point for large (though not near 50%) contamination. However, to attain maximal breakdown, these estimates, like other known maximal breakdown estimators, require large amounts of computational effort. This greater ability of the new estimators to reject outliers comes at a modest cost in efficiency and gross error sensitivity and at a greater, but finite, cost in local shift sensitivity.

Robust State Estimation for Uncertain Systems with Rejection of Harmonic Disturbances

This paper considers a robust state estimation problem for a class of uncertain dynamical systems subject to harmonic disturbances of a known frequency. The class of uncertain systems being considered contain uncertainties described by an Integrad Quadratic Constraint. The state estimator obtained is in the form of a robust Kalman Filter which is constructed by solving a Riccati differential equation of the game type.

Recursive state estimation for uncertain systems with an integral quadratic constraint

This paper considers a robust state estimation problem for a class of uncertain systems where the noise and uncertainty are modeled deterministically via an integral quadratic constraint. The robust state estimation problem involves constructing the set of all possible states at the current time consistent with given output measurements and the integral quadratic constraint.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A game theory approach to robust discrete-time H/sub ∞/-estimation

The deterministic robust estimation problem is formulated in the discrete-time case as a zero-sum two person difference game, in which a statistician plays against nature. The statistician tries to estimate a linear combination of the states of an uncertain linear system, which is driven by nature. Saddle-point strategies for the statistician in the above game are difficult to find. A suboptimal strategy is, therefore, considered which guarantees an H/sub /spl infin// performance bound on the estimation error. Different patterns of the information that is available to the statistician are treated, and the corresponding fixed-point, fixed-lag, and fixed-interval smoothing strategies are derived. >

Integrated models of signal and background with application to speaker identification in noise

Discusses the problem of robust parametric model estimation and classification in noisy acoustic environments. Characterization and modeling of the external noise sources in these environments is in itself an important issue in noise compensation. The techniques described provide a mechanism for integrating parametric models of the acoustic background with the signal model so that noise compensation is tightly coupled with signal model training and classification. Prior information about the acoustic background process is provided using a maximum likelihood parameter estimation procedure that integrates an a priori model of the acoustic background with the signal model. An experimental study is presented on the application of this approach to text-independent speaker identification in noisy acoustic environments. Considerable improvement in speaker classification performance was obtained for classifying unlabeled sections of conversational speech utterances from a 16-speaker population under cross-environment training and testing conditions. >

Least absolute values estimation: computational aspects

A problem that often arises in system identification, the problem of robust parametric estimation, is addressed. A high efficient technique is developed for the recurrent LAV estimation. Due to the direct minimization algorithm, the proposed technique requires half as many independent variables as compared to the simplex method of linear programming, most commonly applied for the LAV estimation. The computational realization is based on the finite iterative scheme. >

The feasible solution algorithm for the minimum covariance determinant estimator in multivariate data

Outliers in multivariate data present a particularly intractable problem. There are various criteria for their identification, and for the related problem of high breakdown robust estimation of multivariate location and scale, but to date no reliable algorithm has been proposed for fitting by these criteria. The most widely used multivariate high breakdown method is the minimum volume ellipsoid (MVE) approach which seeks the ellipsoid of minimum volume containing at least half of the data. The minimum covariance determinant (MCD) approach uses instead the sample mean vector and covariance matrix of a fixed-size subset of the data, the cases in the subset being chosen to minimize the determinant. The MCD approach has better theoretical properties than the MVE, but appears not to be widely used at present, partly no doubt for the lack of an accepted algorithm for finding it. In this paper, we present an algorithm for obtaining the MCD estimator for a given data set. The algorithm is probabilistic; it involves taking random starting ‘trial solutions’ and refining each to a local optimum satisfying the necessary condition for the MCD optimum. The method's convergence is guaranteed only asymptotically as the number of random starting values goes to infinity. We demonstrate however using simulated data and examples of real data from the literature that its practical performance is excellent: that in these data sets it reaches the global optimum with high probability using a very modest number of random starting points. This gives reason to expect that it will handle other real data sets equally effectively, providing a easily computed high breakdown multivariate estimator for use, both in its own right, and as a starting point for other robust procedures.

Bias Robust Estimation of Scale

In this paper we consider the problem of robust estimation of the scale of the location residuals when the underlying distribution of the data belongs to a contamination neighborhood of a parametric location-scale family. We define the class of $M$-estimates of scale with general location, and show that under certain regularity assumptions, these scale estimates converge to their asymptotic functionals uniformly with respect to the underlying distribution, and with respect to the $M$-estimate defining score function $\chi$. We establish expressions for the maximum asymptotic bias of $M$-estimates of scale over the contamination neighborhood as a function of the fraction of contamination. Using these expressions we construct asymptotically min-max bias robust estimates of scale. In particular, we show that a scaled version of the Madm (median of absolute residuals about the median) is approximately min-max bias-robust within the class of Huber's Proposal 2 joint estimates of location and scale. We also consider the larger class of $M$-estimates of scale with general location, and show that a scaled version of the Shorth (the shortest half of the data) is approximately min-max bias robust in this class. Finally, we present the results of a Monte Carlo study showing that the Shorth has attractive finite sample size mean squared error properties for contaminated Gaussian data.

A highly robust estimator through partially likelihood function modeling and its application in computer vision

The authors present a highly robust estimator, known as the model fitting (MF) estimator for general regression. They explain that high robustness becomes possible through partially but completely modeling the unknown log likelihood function. The partial modeling takes place by taking the Bayesian statistical decision rule and a number of important heuristics into consideration while maximizing the log likelihood function. Applications include the automatic selection of multiple thresholds, single rigid motion estimation or multiple rigid motion segmentation, and estimation from two perspective views. It is believed that the proposed MF estimator will aid in solving many robust estimation problems that demand an estimator that is either highly robust or capable of handling contaminated Gaussian mixture models. >

An RKHS approach to robust L/sup 2/ estimation and signal detection

The application of RKHS (reproducing kernel Hilbert space) theory to the problems of robust signal detection and estimation is investigated. It is shown that this approach provides a general and unified framework in which to analyze the problems of L/sup 2/ estimation, matched filtering, and quadratic detection in the presence of uncertainties regarding the second-order structure of the random processes involved. >